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-revision 1.3 by adcroft,
Thu Aug  9 16:26:43 2001 UTC
+revision 1.9 by cnh,
Wed Oct 24 15:21:27 2001 UTC
 Line 8 
 method. This amounts to a grid-point met
  centered finite difference) in the fluid interior but allows
  boundaries to intersect a regular grid allowing a more accurate
  representation of the position of the boundary. We treat the
- horizontal and veritical directions as seperable and thus slightly
+ horizontal and veritical directions as seperable and differently.
- differently.
- Initialization of grid data is controlled by subroutine {\em
+ \input{part2/notation}
- INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the
- vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em
- INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to
- initialize the horizontal grid for cartesian, spherical-polar or
- curvilinear coordinates respectively.
- The reciprocals of all grid quantities are pre-calculated and this is
- done in subroutine {\em INI\_MASKS\_ETC} which is called later by
- subroutine {\em INITIALIZE\_FIXED}.
- All grid descriptors are global arrays and stored in common blocks in
- {\em GRID.h} and a generally declared as {\em \_RS}.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})
- {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
- grid data: ({\em model/inc/GRID.h})
- \end{minipage} }
  \subsection{The finite volume method: finite volumes versus finite difference}
  The finite volume method is used to discretize the equations in
  space. The expression ``finite volume'' actually has two meanings; one
- is the method of cut or instecting boundaries (shaved or lopped cells
+ is the method of embedded or intersecting boundaries (shaved or lopped
- in our terminology) and the other is non-linear interpolation methods
+ cells in our terminology) and the other is non-linear interpolation
- that can deal with non-smooth solutions such as shocks (i.e. flux
+ methods that can deal with non-smooth solutions such as shocks
- limiters for advection). Both make use of the integral form of the
+ (i.e. flux limiters for advection). Both make use of the integral form
- conservation laws to which the {\it weak solution} is a solution on
+ of the conservation laws to which the {\it weak solution} is a
- each finite volume of (sub-domain). The weak solution can be
+ solution on each finite volume of (sub-domain). The weak solution can
- constructed outof piece-wise constant elements or be
+ be constructed out of piece-wise constant elements or be
  differentiable. The differentiable equations can not be satisfied by
  piece-wise constant functions.
-Line 88 
 directions simultaneously. Illustration
+Line 67 
 directions simultaneously. Illustration
  \subsection{C grid staggering of variables}
  \begin{figure}
- \centerline{ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} }
+ \begin{center}
+ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}}
+ \end{center}
  \caption{Three dimensional staggering of velocity components. This
  facilitates the natural discretization of the continuity and tracer
  equations. }
-Line 108 
 synonymous with tracer cells (they are o
+Line 89 
 synonymous with tracer cells (they are o
+ \subsection{Grid initialization and data}
+ Initialization of grid data is controlled by subroutine {\em
+ INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the
+ vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em
+ INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to
+ initialize the horizontal grid for cartesian, spherical-polar or
+ curvilinear coordinates respectively.
+ The reciprocals of all grid quantities are pre-calculated and this is
+ done in subroutine {\em INI\_MASKS\_ETC} which is called later by
+ subroutine {\em INITIALIZE\_FIXED}.
+ All grid descriptors are global arrays and stored in common blocks in
+ {\em GRID.h} and a generally declared as {\em \_RS}.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})
+ {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
+ grid data: ({\em model/inc/GRID.h})
+ \end{minipage} }
  \subsection{Horizontal grid}
+ \label{sec:spatial_discrete_horizontal_grid}
  \begin{figure}
- \centerline{ \begin{tabular}{cc}
+ \begin{center}
+ \begin{tabular}{cc}
    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
  \\
    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
- \end{tabular} }
+ \end{tabular}
+ \end{center}
  \caption{
  Staggering of horizontal grid descriptors (lengths and areas). The
  grid lines indicate the tracer cell boundaries and are the reference
-Line 310 
 other grids, the horizontal grid descrip
+Line 318 
 other grids, the horizontal grid descrip
  \subsection{Vertical grid}
  \begin{figure}
- \centerline{ \begin{tabular}{cc}
+ \begin{center}
+   \begin{tabular}{cc}
    \raisebox{4in}{a)} \resizebox{!}{4in}{
    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}
    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
- \end{tabular} }
+ \end{tabular}
+ \end{center}
  \caption{Two versions of the vertical grid. a) The cell centered
  approach where the interface depths are specified and the tracer
  points centered in between the interfaces. b) The interface centered
-Line 379 
 $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
+Line 389 
 $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
  \subsection{Topography: partially filled cells}
  \begin{figure}
- \centerline{
+ \begin{center}
  \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}
- }
+ \end{center}
  \caption{
  A schematic of the x-r plane showing the location of the
  non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a
-Line 446 
 $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
+Line 456 
 $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
  \end{minipage} }
- \subsection{Continuity and horizontal pressure gradient terms}
+ \section{Continuity and horizontal pressure gradient terms}
  The core algorithm is based on the ``C grid'' discretization of the
  continuity equation which can be summarized as:
  \begin{eqnarray}
- \partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}' & = & G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h' \\
+ \partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}' & = & G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h' \label{eq:discrete-momu} \\
- \partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}' & = & G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h' \\
+ \partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}' & = & G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h' \label{eq:discrete-momv} \\
- \epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right) & = & \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}' \\
+ \epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right) & = & \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}' \label{eq:discrete-momw} \\
  \delta_i \Delta y_g \Delta r_f h_w u +
  \delta_j \Delta x_g \Delta r_f h_s v +
  \delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0}
-Line 476 
 A}_c$.  The factors $h_w$ and $h_s$ are
+Line 486 
 A}_c$.  The factors $h_w$ and $h_s$ are
  The last equation, the discrete continuity equation, can be summed in
  the vertical to yeild the free-surface equation:
  \begin{equation}
- {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v =
+ {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w
- {\cal A}_c(P-E)_{r=0}
+ u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal
+ A}_c(P-E)_{r=0} \label{eq:discrete-freesurface}
  \end{equation}
  The source term $P-E$ on the rhs of continuity accounts for the local
  addition of volume due to excess precipitation and run-off over
  evaporation and only enters the top-level of the {\em ocean} model.
- \subsection{Hydrostatic balance}
+ \section{Hydrostatic balance}
  The vertical momentum equation has the hydrostatic or
  quasi-hydrostatic balance on the right hand side. This discretization
-Line 497 
 discretized:
+Line 508 
 discretized:
  \begin{equation}
  \epsilon_{nh} \partial_t w
  + g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots
+ \label{eq:discrete_hydro_ocean}
  \end{equation}
  In the atmosphere, using p-coordinates, hydrostatic balance is
  discretized:
  \begin{equation}
  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0
+ \label{eq:discrete_hydro_atmos}
  \end{equation}
  where $\Delta \Pi$ is the difference in Exner function between the
  pressure points. The non-hydrostatic equations are not available in
-Line 526 
 CALC\_PHI\_HYD}. Inside this routine, on
+Line 539 
 CALC\_PHI\_HYD}. Inside this routine, on
  atmospheric/oceanic form is selected based on the string variable {\bf
  buoyancyRelation}.
- \subsection{Flux-form momentum equations}
- The original finite volume model was based on the Eulerian flux form
- momentum equations. This is the default though the vector invariant
- form is optionally available (and recommended in some cases).
- The ``G's'' (our colloquial name for all terms on rhs!) are broken
- into the various advective, Coriolis, horizontal dissipation, vertical
- dissipation and metric forces:
- \marginpar{$G_u$: {\bf Gu} }
- \marginpar{$G_v$: {\bf Gv} }
- \marginpar{$G_w$: {\bf Gw} }
- \begin{eqnarray}
- G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} +
- G_u^{metric} + G_u^{nh-metric} \\
- G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +
- G_v^{metric} + G_v^{nh-metric} \\
- G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +
- G_w^{metric} + G_w^{nh-metric}
- \end{eqnarray}
- In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the
- vertical momentum to hydrostatic balance.
- These terms are calculated in routines called from subroutine {\em
- CALC\_MOM\_RHS} a collected into the global arrays {\bf Gu}, {\bf Gv},
- and {\bf Gw}.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_fluxform/calc\_mom\_rhs.F})
- $G_u$: {\bf Gu} ({\em DYNVARS.h})
- $G_v$: {\bf Gv} ({\em DYNVARS.h})
- $G_w$: {\bf Gw} ({\em DYNVARS.h})
- \end{minipage} }
- \subsubsection{Advection of momentum}
- The advective operator is second order accurate in space:
- \begin{eqnarray}
- {\cal A}_w \Delta r_f h_w G_u^{adv} & = &
-   \delta_i \overline{ U }^i \overline{ u }^i
- + \delta_j \overline{ V }^i \overline{ u }^j
- + \delta_k \overline{ W }^i \overline{ u }^k \\
- {\cal A}_s \Delta r_f h_s G_v^{adv} & = &
-   \delta_i \overline{ U }^j \overline{ v }^i
- + \delta_j \overline{ V }^j \overline{ v }^j
- + \delta_k \overline{ W }^j \overline{ v }^k \\
- {\cal A}_c \Delta r_c G_w^{adv} & = &
-   \delta_i \overline{ U }^k \overline{ w }^i
- + \delta_j \overline{ V }^k \overline{ w }^j
- + \delta_k \overline{ W }^k \overline{ w }^k \\
- \end{eqnarray}
- and because of the flux form does not contribute to the global budget
- of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes
- defined:
- \marginpar{$U$: {\bf uTrans} }
- \marginpar{$V$: {\bf vTrans} }
- \marginpar{$W$: {\bf rTrans} }
- \begin{eqnarray}
- U & = & \Delta y_g \Delta r_f h_w u \\
- V & = & \Delta x_g \Delta r_f h_s v \\
- W & = & {\cal A}_c w
- \end{eqnarray}
- The advection of momentum takes the same form as the advection of
- tracers but by a translated advective flow. Consequently, the
- conservation of second moments, derived for tracers later, applies to
- $u^2$ and $v^2$ and $w^2$ so that advection of momentum correctly
- conserves kinetic energy.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_ADV\_UU} ({\em mom\_u\_adv\_uu.F})
- {\em S/R MOM\_U\_ADV\_VU} ({\em mom\_u\_adv\_vu.F})
- {\em S/R MOM\_U\_ADV\_WU} ({\em mom\_u\_adv\_wu.F})
- {\em S/R MOM\_U\_ADV\_UV} ({\em mom\_u\_adv\_uv.F})
- {\em S/R MOM\_U\_ADV\_VV} ({\em mom\_u\_adv\_vv.F})
- {\em S/R MOM\_U\_ADV\_WV} ({\em mom\_u\_adv\_wv.F})
- $uu$, $uv$, $vu$, $vv$: {\bf aF} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Coriolis terms}
- The ``pure C grid'' Coriolis terms (i.e. in absence of C-D scheme) are
- discretized:
- \begin{eqnarray}
- {\cal A}_w \Delta r_f h_w G_u^{Cor} & = &
-   \overline{ f {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i
- - \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ w }^k }^i \\
- {\cal A}_s \Delta r_f h_s G_v^{Cor} & = &
- - \overline{ f {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
- {\cal A}_c \Delta r_c G_w^{Cor} & = &
-  \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ u }^i }^k
- \end{eqnarray}
- where the Coriolis parameters $f$ and $f'$ are defined:
- \begin{eqnarray}
- f & = & 2 \Omega \sin{\phi} \\
- f' & = & 2 \Omega \cos{\phi}
- \end{eqnarray}
- when using spherical geometry, otherwise the $\beta$-plane definition is used:
- \begin{eqnarray}
- f & = & f_o + \beta y \\
- f' & = & 0
- \end{eqnarray}
- This discretization globally conserves kinetic energy. It should be
- noted that despite the use of this discretization in former
- publications, all calculations to date have used the following
- different discretization:
- \begin{eqnarray}
- G_u^{Cor} & = &
-   f_u \overline{ v }^{ji}
- - \epsilon_{nh} f_u' \overline{ w }^{ik} \\
- G_v^{Cor} & = &
- - f_v \overline{ u }^{ij} \\
- G_w^{Cor} & = &
-  \epsilon_{nh} f_w' \overline{ u }^{ik}
- \end{eqnarray}
- \marginpar{Need to change the default in code to match this}
- where the subscripts on $f$ and $f'$ indicate evaluation of the
- Coriolis parameters at the appropriate points in space. The above
- discretization does {\em not} conserve anything, especially energy. An
- option to recover this discretization has been retained for backward
- compatibility testing (set run-time logical {\bf
- useNonconservingCoriolis} to {\em true} which otherwise defaults to
- {\em false}).
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F})
- {\em S/R MOM\_U\_CORIOLIS} ({\em mom\_u\_coriolis.F})
- {\em S/R MOM\_V\_CORIOLIS} ({\em mom\_v\_coriolis.F})
- $G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Curvature metric terms}
- The most commonly used coordinate system on the sphere is the
- geographic system $(\lambda,\phi)$. The curvilinear nature of these
- coordinates on the sphere lead to some ``metric'' terms in the
- component momentum equations. Under the thin-atmosphere and
- hydrostatic approximations these terms are discretized:
- \begin{eqnarray}
- {\cal A}_w \Delta r_f h_w G_u^{metric} & = &
-   \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\
- {\cal A}_s \Delta r_f h_s G_v^{metric} & = &
- - \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
- G_w^{metric} & = & 0
- \end{eqnarray}
- where $a$ is the radius of the planet (sphericity is assumed) or the
- radial distance of the particle (i.e. a function of height).  It is
- easy to see that this discretization satisfies all the properties of
- the discrete Coriolis terms since the metric factor $\frac{u}{a}
- \tan{\phi}$ can be viewed as a modification of the vertical Coriolis
- parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$.
- However, as for the Coriolis terms, a non-energy conserving form has
- exclusively been used to date:
- \begin{eqnarray}
- G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\phi} \\
- G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\phi}
- \end{eqnarray}
- where $\tan{\phi}$ is evaluated at the $u$ and $v$ points
- respectively.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_metric\_sphere.F})
- {\em S/R MOM\_V\_METRIC\_SPHERE} ({\em mom\_v\_metric\_sphere.F})
- $G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Non-hydrostatic metric terms}
- For the non-hydrostatic equations, dropping the thin-atmosphere
- approximation re-introduces metric terms involving $w$ and are
- required to conserve anglular momentum:
- \begin{eqnarray}
- {\cal A}_w \Delta r_f h_w G_u^{metric} & = &
- - \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\
- {\cal A}_s \Delta r_f h_s G_v^{metric} & = &
- - \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j \\
- {\cal A}_c \Delta r_c G_w^{metric} & = &
-   \overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k
- \end{eqnarray}
- Because we are always consistent, even if consistently wrong, we have,
- in the past, used a different discretization in the model which is:
- \begin{eqnarray}
- G_u^{metric} & = &
- - \frac{u}{a} \overline{w}^{ik} \\
- G_v^{metric} & = &
- - \frac{v}{a} \overline{w}^{jk} \\
- G_w^{metric} & = &
-   \frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )
- \end{eqnarray}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metric\_nh.F})
- {\em S/R MOM\_V\_METRIC\_NH} ({\em mom\_v\_metric\_nh.F})
- $G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Lateral dissipation}
- Historically, we have represented the SGS Reynolds stresses as simply
- down gradient momentum fluxes, ignoring constraints on the stress
- tensor such as symmetry.
- \begin{eqnarray}
- {\cal A}_w \Delta r_f h_w G_u^{h-diss} & = &
-   \delta_i  \Delta y_f \Delta r_f h_c \tau_{11}
- + \delta_j  \Delta x_v \Delta r_f h_\zeta \tau_{12} \\
- {\cal A}_s \Delta r_f h_s G_v^{h-diss} & = &
-   \delta_i  \Delta y_u \Delta r_f h_\zeta \tau_{21}
- + \delta_j  \Delta x_f \Delta r_f h_c \tau_{22}
- \end{eqnarray}
- \marginpar{Check signs of stress definitions}
- The lateral viscous stresses are discretized:
- \begin{eqnarray}
- \tau_{11} & = & A_h c_{11\Delta}(\phi) \frac{1}{\Delta x_f} \delta_i u
-                -A_4 c_{11\Delta^2}(\phi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\
- \tau_{12} & = & A_h c_{12\Delta}(\phi) \frac{1}{\Delta y_u} \delta_j u
-                -A_4 c_{12\Delta^2}(\phi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\
- \tau_{21} & = & A_h c_{21\Delta}(\phi) \frac{1}{\Delta x_v} \delta_i v
-                -A_4 c_{21\Delta^2}(\phi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\
- \tau_{22} & = & A_h c_{22\Delta}(\phi) \frac{1}{\Delta y_f} \delta_j v
-                -A_4 c_{22\Delta^2}(\phi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v
- \end{eqnarray}
- where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in
- \{1,2\}$ define the ``cosine'' scaling with latitude which can be
- applied in various ad-hoc ways. For instance, $c_{11\Delta} =
- c_{21\Delta} = (\cos{\phi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would
- represent the an-isotropic cosine scaling typically used on the
- ``lat-lon'' grid for Laplacian viscosity.
- \marginpar{Need to tidy up method for controlling this in code}
- It should be noted that dispite the ad-hoc nature of the scaling, some
- scaling must be done since on a lat-lon grid the converging meridians
- make it very unlikely that a stable viscosity parameter exists across
- the entire model domain.
- The Laplacian viscosity coefficient, $A_h$ ({\bf viscAh}), has units
- of $m^2 s^{-1}$. The bi-harmonic viscosity coefficient, $A_4$ ({\bf
- viscA4}), has units of $m^4 s^{-1}$.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_XVISCFLUX} ({\em mom\_u\_xviscflux.F})
- {\em S/R MOM\_U\_YVISCFLUX} ({\em mom\_u\_yviscflux.F})
- {\em S/R MOM\_V\_XVISCFLUX} ({\em mom\_v\_xviscflux.F})
- {\em S/R MOM\_V\_YVISCFLUX} ({\em mom\_v\_yviscflux.F})
- $\tau_{11}$, $\tau_{12}$, $\tau_{22}$, $\tau_{22}$: {\bf vF}, {\bf
- v4F} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- Two types of lateral boundary condition exist for the lateral viscous
- terms, no-slip and free-slip.
- The free-slip condition is most convenient to code since it is
- equivalent to zero-stress on boundaries. Simple masking of the stress
- components sets them to zero. The fractional open stress is properly
- handled using the lopped cells.
- The no-slip condition defines the normal gradient of a tangential flow
- such that the flow is zero on the boundary. Rather than modify the
- stresses by using complicated functions of the masks and ``ghost''
- points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as
- an additional source term in cells next to solid boundaries. This has
- the advantage of being able to cope with ``thin walls'' and also makes
- the interior stress calculation (code) independent of the boundary
- conditions. The ``body'' force takes the form:
- \begin{eqnarray}
- G_u^{side-drag} & = &
- \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j
- \left( A_h c_{12\Delta}(\phi) u - A_4 c_{12\Delta^2}(\phi) \nabla^2 u \right)
- \\
- G_v^{side-drag} & = &
- \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i
- \left( A_h c_{21\Delta}(\phi) v - A_4 c_{21\Delta^2}(\phi) \nabla^2 v \right)
- \end{eqnarray}
- In fact, the above discretization is not quite complete because it
- assumes that the bathymetry at velocity points is deeper than at
- neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F})
- {\em S/R MOM\_V\_SIDEDRAG} ({\em mom\_v\_sidedrag.F})
- $G_u^{side-drag}$, $G_v^{side-drag}$: {\bf vF} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Vertical dissipation}
- Vertical viscosity terms are discretized with only partial adherence
- to the variable grid lengths introduced by the finite volume
- formulation. This reduces the formal accuracy of these terms to just
- first order but only next to boundaries; exactly where other terms
- appear such as linar and quadratic bottom drag.
- \begin{eqnarray}
- G_u^{v-diss} & = &
- \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
- G_v^{v-diss} & = &
- \frac{1}{\Delta r_f h_s} \delta_k \tau_{23} \\
- G_w^{v-diss} & = & \epsilon_{nh}
- \frac{1}{\Delta r_f h_d} \delta_k \tau_{33}
- \end{eqnarray}
- represents the general discrete form of the vertical dissipation terms.
- In the interior the vertical stresses are discretized:
- \begin{eqnarray}
- \tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\
- \tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v \\
- \tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w
- \end{eqnarray}
- It should be noted that in the non-hydrostatic form, the stress tensor
- is even less consistent than for the hydrostatic (see Wazjowicz
- \cite{Waojz}). It is well known how to do this properly (see Griffies
- \cite{Griffies}) and is on the list of to-do's.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
- {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})
- $\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})
- $\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- As for the lateral viscous terms, the free-slip condition is
- equivalent to simply setting the stress to zero on boundaries.  The
- no-slip condition is implemented as an additional term acting on top
- of the interior and free-slip stresses. Bottom drag represents
- additional friction, in addition to that imposed by the no-slip
- condition at the bottom. The drag is cast as a stress expressed as a
- linear or quadratic function of the mean flow in the layer above the
- topography:
- \begin{eqnarray}
- \tau_{13}^{bottom-drag} & = &
- \left(
-A_v \frac{1}{\Delta r_c}
- + r_b
- + C_d \sqrt{ \overline{2 KE}^i }
- \right) u \\
- \tau_{23}^{bottom-drag} & = &
- \left(
-A_v \frac{1}{\Delta r_c}
- + r_b
- + C_d \sqrt{ \overline{2 KE}^j }
- \right) v
- \end{eqnarray}
- where these terms are only evaluated immediately above topography.
- $r_b$ ({\bf bottomDragLinear}) has units of $m s^{-1}$ and a typical value
- of the order 0.0002 $m s^{-1}$. $C_d$ ({\bf bottomDragQuadratic}) is
- dimensionless with typical values in the range 0.001--0.003.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_BOTTOMDRAG} ({\em mom\_u\_bottomdrag.F})
- {\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F})
- $\tau_{13}^{bottom-drag}$, $\tau_{23}^{bottom-drag}$: {\bf vf} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsection{Tracer equations}
- The tracer equations are discretized consistantly with the continuity
- equation to facilitate conservation properties analogous to the
- continuum:
- \begin{equation}
- {\cal A}_c \Delta r_f h_c \partial_\theta
- + \delta_i U \overline{ \theta }^i
- + \delta_j V \overline{ \theta }^j
- + \delta_k W \overline{ \theta }^k
- = {\cal A}_c \Delta r_f h_c {\cal S}_\theta + \theta {\cal A}_c \delta_k (P-E)_{r=0}
- \end{equation}
- The quantities $U$, $V$ and $W$ are volume fluxes defined:
- \marginpar{$U$: {\bf uTrans} }
- \marginpar{$V$: {\bf vTrans} }
- \marginpar{$W$: {\bf rTrans} }
- \begin{eqnarray}
- U & = & \Delta y_g \Delta r_f h_w u \\
- V & = & \Delta x_g \Delta r_f h_s v \\
- W & = & {\cal A}_c w
- \end{eqnarray}
- ${\cal S}$ represents the ``parameterized'' SGS processes and
- physics associated with the tracer. For instance, potential
- temperature equation in the ocean has is forced by surface and
- partially penetrating heat fluxes:
- \begin{equation}
- {\cal A}_c \Delta r_f h_c {\cal S}_\theta = \frac{1}{c_p \rho_o} \delta_k {\cal A}_c {\cal Q}
- \end{equation}
- while the salt equation has no real sources, ${\cal S}=0$, which
- leaves just the $P-E$ term.
- The continuity equation can be recovered by setting ${\cal Q}=0$ and
- $\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local
- conservation of $\theta$. Global conservation is not possible using
- the flux-form (as here) and a linearized free-surface
- (\cite{Griffies00,Campin02}).
- \subsection{Derivation of discrete energy conservation}
- These discrete equations conserve kinetic plus potential energy using the
- following definitions:
- \begin{equation}
- KE = \frac{1}{2} \left( \overline{ u^2 }^i + \overline{ v^2 }^j +
- \epsilon_{nh} \overline{ w^2 }^k \right)
- \end{equation}
- \subsection{Vector invariant momentum equations}
- The finite volume method lends itself to describing the continuity and
- tracer equations in curvilinear coordinate systems but the appearance
- of new metric terms in the flux-form momentum equations makes
- generalizing them far from elegant. The vector invariant form of the
- momentum equations are exactly that; invariant under coordinate
- transformations.
- The non-hydrostatic vector invariant equations read:
- \begin{equation}
- \partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v}
- - b \hat{r}
- + \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau}
- \end{equation}
- which describe motions in any orthogonal curvilinear coordinate
- system. Here, $B$ is the Bernoulli function and $\vec{\zeta}=\nabla
- \wedge \vec{v}$ is the vorticity vector. We can take advantage of the
- elegance of these equations when discretizing them and use the
- discrete definitions of the grad, curl and divergence operators to
- satisfy constraints. We can also consider the analogy to forming
- derived equations, such as the vorticity equation, and examine how the
- discretization can be adjusted to give suitable vorticity advection
- among other things.
- The underlying algorithm is the same as for the flux form
- equations. All that has changed is the contents of the ``G's''. For
- the time-being, only the hydrostatic terms have been coded but we will
- indicate the points where non-hydrostatic contributions will enter:
- \begin{eqnarray}
- G_u & = & G_u^{fv} + G_u^{\zeta_3 v} + G_u^{\zeta_2 w} + G_u^{\partial_x B}
- + G_u^{\partial_z \tau^x} + G_u^{h-dissip} + G_u^{v-dissip} \\
- G_v & = & G_v^{fu} + G_v^{\zeta_3 u} + G_v^{\zeta_1 w} + G_v^{\partial_y B}
- + G_v^{\partial_z \tau^y} + G_v^{h-dissip} + G_v^{v-dissip} \\
- G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G_w^{\zeta_2 u} + G_w^{\partial_z B}
- + G_w^{h-dissip} + G_w^{v-dissip}
- \end{eqnarray}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/calc\_mom\_rhs.F})
- $G_u$: {\bf Gu} ({\em DYNVARS.h})
- $G_v$: {\bf Gv} ({\em DYNVARS.h})
- $G_w$: {\bf Gw} ({\em DYNVARS.h})
- \end{minipage} }
- \subsubsection{Relative vorticity}
- The vertical component of relative vorticity is explicitly calculated
- and use in the discretization. The particular form is crucial for
- numerical stablility; alternative definitions break the conservation
- properties of the discrete equations.
- Relative vorticity is defined:
- \begin{equation}
- \zeta_3 = \frac{\Gamma}{A_\zeta}
- = \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c v - \delta_j \Delta x_c u )
- \end{equation}
- where ${\cal A}_\zeta$ is the area of the vorticity cell presented in
- the vertical and $\Gamma$ is the circulation about that cell.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})
- $\zeta_3$: {\bf vort3} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Kinetic energy}
- The kinetic energy, denoted $KE$, is defined:
- \begin{equation}
- KE = \frac{1}{2} ( \overline{ u^2 }^i + \overline{ v^2 }^j
- + \epsilon_{nh} \overline{ w^2 }^k )
- \end{equation}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})
- $KE$: {\bf KE} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Coriolis terms}
- The potential enstrophy conserving form of the linear Coriolis terms
- are written:
- \begin{eqnarray}
- G_u^{fv} & = &
- \frac{1}{\Delta x_c}
- \overline{ \frac{f}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
- G_v^{fu} & = & -
- \frac{1}{\Delta y_c}
- \overline{ \frac{f}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
- \end{eqnarray}
- Here, the Coriolis parameter $f$ is defined at vorticity (corner)
- points.
- \marginpar{$f$: {\bf fCoriG}}
- \marginpar{$h_\zeta$: {\bf hFacZ}}
- The potential enstrophy conserving form of the non-linear Coriolis
- terms are written:
- \begin{eqnarray}
- G_u^{\zeta_3 v} & = &
- \frac{1}{\Delta x_c}
- \overline{ \frac{\zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
- G_v^{\zeta_3 u} & = & -
- \frac{1}{\Delta y_c}
- \overline{ \frac{\zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
- \end{eqnarray}
- \marginpar{$\zeta_3$: {\bf vort3}}
- The Coriolis terms can also be evaluated together and expressed in
- terms of absolute vorticity $f+\zeta_3$. The potential enstrophy
- conserving form using the absolute vorticity is written:
- \begin{eqnarray}
- G_u^{fv} + G_u^{\zeta_3 v} & = &
- \frac{1}{\Delta x_c}
- \overline{ \frac{f + \zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\
- G_v^{fu} + G_v^{\zeta_3 u} & = & -
- \frac{1}{\Delta y_c}
- \overline{ \frac{f + \zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j
- \end{eqnarray}
- \marginpar{Run-time control needs to be added for these options} The
- disctinction between using absolute vorticity or relative vorticity is
- useful when constructing higher order advection schemes; monotone
- advection of relative vorticity behaves differently to monotone
- advection of absolute vorticity. Currently the choice of
- relative/absolute vorticity, centered/upwind/high order advection is
- available only through commented subroutine calls.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_CORIOLIS} ({\em mom\_vi\_coriolis.F})
- {\em S/R MOM\_VI\_U\_CORIOLIS} ({\em mom\_vi\_u\_coriolis.F})
- {\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F})
- $G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})
- $G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Shear terms}
- The shear terms ($\zeta_2w$ and $\zeta_1w$) are are discretized to
- guarantee that no spurious generation of kinetic energy is possible;
- the horizontal gradient of Bernoulli function has to be consistent
- with the vertical advection of shear:
- \marginpar{N-H terms have not been tried!}
- \begin{eqnarray}
- G_u^{\zeta_2 w} & = &
- \frac{1}{ {\cal A}_w \Delta r_f h_w } \overline{
- \overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )
- }^k \\
- G_v^{\zeta_1 w} & = &
- \frac{1}{ {\cal A}_s \Delta r_f h_s } \overline{
- \overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )
- }^k
- \end{eqnarray}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_U\_VERTSHEAR} ({\em mom\_vi\_u\_vertshear.F})
- {\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})
- $G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})
- $G_v^{\zeta_1 w}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Gradient of Bernoulli function}
- \begin{eqnarray}
- G_u^{\partial_x B} & = &
- \frac{1}{\Delta x_c} \delta_i ( \phi' + KE ) \\
- G_v^{\partial_y B} & = &
- \frac{1}{\Delta x_y} \delta_j ( \phi' + KE )
- %G_w^{\partial_z B} & = &
- %\frac{1}{\Delta r_c} h_c \delta_k ( \phi' + KE )
- \end{eqnarray}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_U\_GRAD\_KE} ({\em mom\_vi\_u\_grad\_ke.F})
- {\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})
- $G_u^{\partial_x KE}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})
- $G_v^{\partial_y KE}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Horizontal dissipation}
- The horizontal divergence, a complimentary quantity to relative
- vorticity, is used in parameterizing the Reynolds stresses and is
- discretized:
- \begin{equation}
- D = \frac{1}{{\cal A}_c h_c} (
-   \delta_i \Delta y_g h_w u
- + \delta_j \Delta x_g h_s v )
- \end{equation}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})
- $D$: {\bf hDiv} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Horizontal dissipation}
- The following discretization of horizontal dissipation conserves
- potential vorticity (thickness weighted relative vorticity) and
- divergence and dissipates energy, enstrophy and divergence squared:
- \begin{eqnarray}
- G_u^{h-dissip} & = &
-   \frac{1}{\Delta x_c} \delta_i ( A_D D - A_{D4} D^*)
- - \frac{1}{\Delta y_u h_w} \delta_j h_\zeta ( A_\zeta \zeta - A_{\zeta4} \zeta^* )
- \\
- G_v^{h-dissip} & = &
-   \frac{1}{\Delta x_v h_s} \delta_i h_\zeta ( A_\zeta \zeta - A_\zeta \zeta^* )
- + \frac{1}{\Delta y_c} \delta_j ( A_D D - A_{D4} D^* )
- \end{eqnarray}
- where
- \begin{eqnarray}
- D^* & = & \frac{1}{{\cal A}_c h_c} (
-   \delta_i \Delta y_g h_w \nabla^2 u
- + \delta_j \Delta x_g h_s \nabla^2 v ) \\
- \zeta^* & = & \frac{1}{{\cal A}_\zeta} (
-   \delta_i \Delta y_c \nabla^2 v
- - \delta_j \Delta x_c \nabla^2 u )
- \end{eqnarray}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})
- $G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F})
- $G_v^{h-dissip}$: {\bf vDiss} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }
- \subsubsection{Vertical dissipation}
- Currently, this is exactly the same code as the flux form equations.
- \begin{eqnarray}
- G_u^{v-diss} & = &
- \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\
- G_v^{v-diss} & = &
- \frac{1}{\Delta r_f h_s} \delta_k \tau_{23}
- \end{eqnarray}
- represents the general discrete form of the vertical dissipation terms.
- In the interior the vertical stresses are discretized:
- \begin{eqnarray}
- \tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\
- \tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v
- \end{eqnarray}
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
- {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})
- $\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})
- $\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})
- \end{minipage} }

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